Sample Exam MATH3078 PDEs and Waves
1. (a) Using the method of characteristics, solve the equation
ux + xuy = u for x > 1, y ∈ R,
subject to the initial condition u(1, y) = sin y for y ∈ R.
(b) Consider the following second order linear PDE
4y2 uxx − e2x uyy − 4y2 ux = 0 for x ∈ R and y < 0.
(i) Determine its type on the given domain.
(ii) Determine a suitable change of variables, ξ = ξ(x, y) and η = η(x, y)
which, after putting v(ξ, η) = u(x, y), gives the canonical form of the
transformed equation. Do not bring the equation to its canonical form.
2. (a) Consider the Cauchy problem for the linear one-dimensional wave equation
utt = uxx for x ∈ R and t > 0,
u(x, 0) = f(x) for x ∈ R,
ut(x, 0) = g(x) for x ∈ R,
where f ∈ C2(R) and g ∈ C1(R). Show that if f and g are odd functions,
then for every fixed t > 0, the function u(0, t) is necessarily equal to 0.
(b) (i) Without proving, write down the Laplace equation in polar coordi-
nates and the formula for the general solution of the Laplace equation
in a disk in R2 centred at (0, 0) and of radius
√
6.
(ii) Let D = {(x, y) ∈ R2 : x2 + y2 < 6}. Find a harmonic function u in
the disk D, satisfying u(x, y) = y + y2 on the boundary of D. Write
your answer in a Cartesian coordinate system.
3. Consider the regular Sturm–Liouville eigenvalue problem− (xux)x +
u
x
= λ
u
x
for 1 < x < e,
u(1) = u(e) = 0.
(1)
(a) Determine the eigenvalue problem solved by v(y), where we define
v(y) = u(x) with y = lnx.
(b) Using (a), find all eigenvalues {λn}n≥1 of problem (1) and show that a cor-
responding eigenfunction φn for λn is
φn(x) =
√
2 sin (npi lnx) .
(c) Check directly that the sequence of eigenfunctions {φn}n≥1 of (1) is orthonor-
mal with respect to an appropriate inner product.
Sample Exam MATH3078 PDEs and Waves page 2 of 2
4. (a) Consider the boundary value problem− (xux)x +
u
x
=
1
x
for 1 < x < e,
u(1) = u(e) = 0.
(2)
(i) Justify why problem (2) has at most one C2-solution on [1, e].
(ii) Find the solution to the boundary value problem (2). Hint: Using
the orthonormal sequence {φn}n≥1 of eigenfunctions of (1) from Ques-
tion 3(b), look for a solution of the form u(x) =
∑∞
n=1 cnφn(x) and
determine the coefficients cn using generalized Fourier series.
(b) Let f ∈ C1([1,∞)) satisfy f(1) = 1 and f(e) = 0. Consider the following
non-homogeneous initial boundary value problem
wt + w − x(xwx)x = − e
1− e for 1 < x < e and t > 0,
w(1, t) = 1, w(e, t) = 0 for t > 0,
w(x, 0) = f(x) for 1 ≤ x ≤ e.
(3)
(i) Find the constants a and b such that u(x, t) = w(x, t) + ax+ b solves
the corresponding homogeneous equation
ut + u− x(xux)x = 0 for 1 < x < e and t > 0,
subject to
u(1, t) = u(e, t) = 0 for t > 0.
(ii) Solve the initial boundary value problem corresponding to u(x, t) using
the method of separation of variables and Question 3. (Write the
Fourier coefficients in integral form without further calculation.)
(iii) Hence, find a solution w of (3).
This is the end of the examination paper